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Full text of "Mathematical And Physical Papers - Iii"

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*JO to *JK, where K is constant, and let  77, f, be the results. The equation (5) becomes
du     T7. fdzu    d*u    d
This will be true whatever be the value of K, but it will be convenient to suppose that
KS = ABC ........................ (12).
Now imagine a second solid formed from the first by altering all lines parallel to x in the ratio of */A to *JK, all lines parallel to y in the ratio of *JB to *JK, and all lines parallel to z in the ratio of \/C to \/K, nothing being as yet specified regarding the nature of the second solid, except that it is homogeneous. Imagine any number of points, lines, surfaces, or spaces, conceived as belonging to the first solid, and let the points, lines, &c. deduced from them by altering the coordinates in the ratios above-mentioned, and conceived as belonging to the second solid, be said to correspond to the others. On account of the particular magnitude of K chosen, it is evident that the volumes of corresponding spaces will be equal. Let the second solid be called the auxiliary solid, and the operation of deducing either solid from the other, derivation; arid suppose the temperatures equal at corresponding points of the two solids.
The equation (11) shews that the successive distributions of temperature in the interior of the auxiliary solid will take place as if this solid were an ordinary medium in which the interior conductivity bears to K the same ratio that the product of the specific heat and density bears to cp.
The first of equations (7) gives
If now wo refer fx, not to a unit of surface, but to that area of a plane perpendicular to the axis of x which is changed by derivation into a unit of .surface, wo must multiply the above expression by \J(BO) and divide it by K. Denoting the result by /$ , using /J,, f{ to denote for y, z, what fg denotes for #, and taking account of (12), we get